25(0.25-r+r^2)=47r^2

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Solution for 25(0.25-r+r^2)=47r^2 equation:


Simplifying
25(0.25 + -1r + r2) = 47r2
(0.25 * 25 + -1r * 25 + r2 * 25) = 47r2
(6.25 + -25r + 25r2) = 47r2

Solving
6.25 + -25r + 25r2 = 47r2

Solving for variable 'r'.

Combine like terms: 25r2 + -47r2 = -22r2
6.25 + -25r + -22r2 = 47r2 + -47r2

Combine like terms: 47r2 + -47r2 = 0
6.25 + -25r + -22r2 = 0

Begin completing the square.  Divide all terms by
-22 the coefficient of the squared term: 

Divide each side by '-22'.
-0.2840909091 + 1.136363636r + r2 = 0

Move the constant term to the right:

Add '0.2840909091' to each side of the equation.
-0.2840909091 + 1.136363636r + 0.2840909091 + r2 = 0 + 0.2840909091

Reorder the terms:
-0.2840909091 + 0.2840909091 + 1.136363636r + r2 = 0 + 0.2840909091

Combine like terms: -0.2840909091 + 0.2840909091 = 0.0000000000
0.0000000000 + 1.136363636r + r2 = 0 + 0.2840909091
1.136363636r + r2 = 0 + 0.2840909091

Combine like terms: 0 + 0.2840909091 = 0.2840909091
1.136363636r + r2 = 0.2840909091

The r term is 1.136363636r.  Take half its coefficient (0.568181818).
Square it (0.3228305783) and add it to both sides.

Add '0.3228305783' to each side of the equation.
1.136363636r + 0.3228305783 + r2 = 0.2840909091 + 0.3228305783

Reorder the terms:
0.3228305783 + 1.136363636r + r2 = 0.2840909091 + 0.3228305783

Combine like terms: 0.2840909091 + 0.3228305783 = 0.6069214874
0.3228305783 + 1.136363636r + r2 = 0.6069214874

Factor a perfect square on the left side:
(r + 0.568181818)(r + 0.568181818) = 0.6069214874

Calculate the square root of the right side: 0.779051659

Break this problem into two subproblems by setting 
(r + 0.568181818) equal to 0.779051659 and -0.779051659.

Subproblem 1

r + 0.568181818 = 0.779051659 Simplifying r + 0.568181818 = 0.779051659 Reorder the terms: 0.568181818 + r = 0.779051659 Solving 0.568181818 + r = 0.779051659 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.568181818' to each side of the equation. 0.568181818 + -0.568181818 + r = 0.779051659 + -0.568181818 Combine like terms: 0.568181818 + -0.568181818 = 0.000000000 0.000000000 + r = 0.779051659 + -0.568181818 r = 0.779051659 + -0.568181818 Combine like terms: 0.779051659 + -0.568181818 = 0.210869841 r = 0.210869841 Simplifying r = 0.210869841

Subproblem 2

r + 0.568181818 = -0.779051659 Simplifying r + 0.568181818 = -0.779051659 Reorder the terms: 0.568181818 + r = -0.779051659 Solving 0.568181818 + r = -0.779051659 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.568181818' to each side of the equation. 0.568181818 + -0.568181818 + r = -0.779051659 + -0.568181818 Combine like terms: 0.568181818 + -0.568181818 = 0.000000000 0.000000000 + r = -0.779051659 + -0.568181818 r = -0.779051659 + -0.568181818 Combine like terms: -0.779051659 + -0.568181818 = -1.347233477 r = -1.347233477 Simplifying r = -1.347233477

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.210869841, -1.347233477}

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